Annulus Point Vortex Dynamics

For the Circular annulus case we know the closed form solution for the evolution for the single point vortex dynamics so we can verify our mechanics for that case.

Exterior Calculus Derivation

#TODO

Analytical Derivation for Harmonic Part

For a circular annulus

$$A = \{ x \in \mathbb{R}^{2} : a < \lvert x \rvert\ < b \}$$

with one point vortex on the plane represented as:

$$x(t) = r(t)\,e^{i\theta(t)}$$

where $a < r(t) < b$ is the distance form the origin and $\theta(t)$ represents the angle with the x-axis.

For an annulus, we can think of the stream function coming from an imaginary point vortex kept at the center of the annulus. We don’t know the strength for. For such point vortex, the stream function is given as:

$$\psi_{h} = \frac{\Gamma}{2\pi}\log r\,+\,C \implies A\log r + C$$

where $A$ and $C$ are constants. Based on boundary values:

$$A\log r_{in} + C = 1$$ $$A\log r_{out} + C = 0$$ $$A\log r_{in} - A\log r_{out} = 1$$ $$A = \frac{1}{\log\left( r_{in}/r_{out} \right)}$$ $$C = \frac{\log(r_{out})}{\log(r_{in}/r_{out})}$$ $$\therefore \boxed{\psi_{h}= \frac{\log(r/r_{out})}{\log(r_{in}/r_{out})}}$$

Then,

$$\nabla \psi_{h} = \frac{1}{\log(r_{in}/r_{out})} \frac{1}{r} \mathbf{e}_{r}$$

Also, $J\mathbf{e}_{r}=\mathbf{e_{\theta}}$

$$\therefore J\nabla \psi_{h}(r) = \boxed{u_{h}(r)= \frac{1}{r\log(r_{in}/r_{out})} \mathbf{e_{\theta}}}$$

taking circulation around a loop around the center hole of the annulus:

$$\kappa = \oint_{\gamma}u_{h}(r) \cdot dl$$

$d\vec{l} = r\,d\theta\,\mathbf{e_{\theta}}$, then

$$\kappa = \oint_{y} \frac{1}{r\log(r_{in}/r_{out})} r\,d\theta \implies \boxed{\kappa = \frac{2\pi}{\log(r_{in}/r_{out})}}$$

Analytical Derivation of Co-Exact Part

$$z = r e^{i\theta} \to \log(z) \to w = x + i y \to \log r + i\theta\to$$ $$\text{circle}\,r \to R \to \text{line}\,x = \log R$$ $$r \mapsto \frac{R^{2}}{r} \to x \mapsto 2\log R - x$$

This log transformation makes the inversion as just a plane mirror reflection.

https://chatgpt.com/share/69eecc63-23dc-83e8-a5e2-3f1ab9f3f6b2

• Generating Lattice from images by the conformal map.
• Finding a theta function
• Stream function from complex potential
• Removing self-singularity using mean value property
• Velocity from stream function
  • Wirtinger Derivatives
  • Velocity calculation

Simulation Result Analysis

• Visualize the stream function - Need anyways
  • What’s the stream function
• Matching stream function values to the Poisson solve case
• Today:
  • Velocity visualization away from vortex
  • Velocity at the core comparison DEC vs analytical case.
  • Stream-function values at the boundary.
• Validations
  • Rotational In-variance: Velocity constant for R no change over $\theta$.
  • Trajectory validation: